Math A-100 IUSB Sec 1.7 detailed notes – Week 2 Exponents Exponents are NOT real numbers – they are simply a short-hand way of writing out longer multiplication questions. There are two parts to each number with an exponent: the base and the exponent 54 the base is 5 and the exponent is 4 For example: Exponents only apply to whatever is immediately to the left of the exponent itself, i.e. if the exponent is next to a ( ) then it applies to the entire ( ), but if the exponent is next to just the number, then only the number it repeated. Example 1: Compared to: (-5)4  (-5)*(-5)*(-5)*(-5) = 625 -54  -5*5*5*5 = -625 NOTE: the exponent is next to the 5 (not the “-“ sign), so only the 5 gets repeated Example 2: Compared to: Example 3: (2/3)4 (2/3)*( 2/3)*( 2/3)*( 2/3) = 16/81 24  2*2*2*2 = 16 3 3 3 4x6 means 4*x*x*x*x*x*x Compared to: (4x)6 means 4x*4x*4x*4x*4x*4x When 0 is the Exponent When 0 is the exponent, you have a special case, because the answer is 1: Example 1: 70 = ?? Let’s walk it down from a larger exponent to see why the answer is 1 74 = 7*7*7*7 to go from 74 to 73, divide by 7, giving you: 73 = 7*7*7 to go from 73 to 72, divide by 7, giving you 72 = 7*7 to go from 72 to 71, divide by 7, giving you 71 = 7 to go from 71 to 70, divide by 7, giving you 70 = 1 It also works with variables. Example 1: x0 = ?? Following the method used in Example 1: x4 = x*x*x*x to go from x4 to x3, divide by x, giving you: x3 = x*x*x to go from x3 to x2, divide by x, giving you x2 = x*x to go from x2 to x1, divide by x, giving you x1 = x to go from x1 to x0, divide by x, giving you x0 = 1 Exponents outside the ( ) If you have an exponent outside of the ( ), the exponent applies to everything inside the ( ). Think of it as ___ sets of each number inside the ( ). Then you can move the numbers with the negative exponents to make the exponents positive, and simplify. Steps: a) b) c) d) Apply the outside exponent individually to each number inside the ( ) Calculate how many of each of the numbers you now have Move any numbers with negative exponents to make the exponents positive Simplify the fraction Example 1: 53 26 4 a) (53)4 (26)4 b) 4 sets of 3 = 12 and 4 sets of 6 = 24, giving you 512 224 c) Move the numbers to change the negative exponents into positive exponents 224 512 Example 2: write with positive exponents only: a) (x7)3 x7 y9 3 (y9)3 b) 3 sets of 7 = 21 and 3 sets of 9 = 27, giving you x21 y27 Dividing numbers with exponents When you are given numbers in exponents and asked to divide them: 1) Try it this way: a. Write out the number showing how many of each you have, according to the exponent b. Cancel out the exponents into one single power of 10 Example 1: 107 105 (107) = 10*10*10*10*10*10*10 = 102 (105) 10*10*10*10*10 Example 2: 68 62 62 = ____6*6________ = 1 68 6*6*6*6*6*6*6*6 66 Example 3: x9 x5 x9 = x*x*x*x*x*x*x*x*x = x4 x5 x*x*x*x*x NOTE: Did you notice the short-cut? Cancelling out the numbers in the top and bottom is the same as subtracting them. Order of operations with exponents, solving equations involving exponents As noted before, there is no such thing as the Order of Operations, because there is no 1 set order that works in every question. Each question must be taken on a case-by-case basis. Forget all about PEMDAS, because it is a lie. Again, some things to think about/why we handle some things early on: Exponents – they are simply short-hand, numbers with exponent are NOT REAL numbers, so we have to find out what the REAL number is Parentheses – they are like a “bag” containing a single value. We have to find out what that single value is BEFORE we can go on Multiplying – this represents “groups of ____”. We need to find out how many total pieces you have Dividing – this represents breaking something up into smaller groups, so we need to find out how many are in each smaller group before we go on Adding – we need to know how many we have before we can add more to it Subtracting - we need to know how many we have before we can take some away from it. NOTE: Brackets, [ ], mean the same thing as parentheses, ( ), i.e. a “bag”. They are sometimes both used in the same question to help you keep things straight, because 3(7 - 8(6 – (-9))) can get pretty confusing. Example 1: simplify: (3/4)2 * (5/6)2 (3/4)2 * (5/6)2 Do exponent first, because they are not real numbers, giving you: 9/16 * 25/36 then do the multiplying, giving you 225/576 which does not reduce Example 2: simplify: (1/3 + 1/4)*(1/5) + (1/2)3 (1/3 + 1/4)*(1/5) + (1/2)3 Do exponent first (not real numbers), giving you: (1/3 + 1/4)*(1/5) + 1/8 Do the ( ) with the +, as it represents 1 single value (7/12)*(1/5) + 1/8 Do the *, as it represent “groups” 7/60 + 1/8 finish it 7/60 + 1/8  21/240 + 30/240  51/240 which reduces to 17/80 Example 3: simplify: 3 3 - 4 2 33 - 42 62 – (-8) 2 Do exponents first (not real numbers), giving you: 62 – (-8) 2 Do “-“ on top & on bottom, as they represent a single value each 27 - 16 36 – 64 11 -28 which simplifies to _ 11 28 Solving Equations You will get a wide variety of different looking equations and ask you to solve for the variable, usually “x”. My best suggestion is to keep the following phrase in mind: Undo what they gave you: Example 1: 3x = 18 They gave you “3 multiplied by x” so use division to undo the multiplication, i.e. divide both sides by 3 3x/3 = 18/3 x=6 Example 2: x - 3 = 18 They gave you “x subtract 3” so use addition to undo the subtraction, i.e. add 3 to both sides: x – 3 + 3 = 18 + 3 x = 21 Example 3: sqrt(x) = 8 They gave you “square root of x” so square both sides to undo the square root: [Sqrt(x)]2 = [8]2 x = 64 Example 4: x2 = 49 They gave you “x squared so take the square root of both sides to undo the square: Sqrt(x2) = sqrt[49] Example 5: x=7 3x = 18 4 They gave you “3/4 multiplied by x”. When working with fractions, you have a few options: a) use division to undo the multiplication, i.e. divide both sides by 3/4 3x divide by 3 = 18 divided by 3 4 4 4 x = 18 divided by 3 4 giving you: flip to multiplication & flip the fraction: x = 18 * 4 which gives you: 3 x = 24 b) since you end up flipping the fraction and multiplying anyway, just go there directly and skip the first part: 3x * 4 = 18 * 4 4 3 3 x = 18 * 4 3 x = 24 giving you: which gives you: c) break it up and do it in parts: 3x = 18 4 They gave you “3 multiplied by x” & “x divided by 4”, so undo each one separately Multiply by 4 to undo the dividing by 4 3x * 4 = 18 * 4 4 which gives you: 3x = 72 then divide both sides by 3 to undo the multiplying: 3x/3 = 72/3 which gives you x = 24 Example 6: .4x = 18 Decimals are just like whole numbers. they gave you “.4 multiplied by x” so use division to undo the multiplication, i.e. divide both sides by .4 .4x/.4 = 18/.4 x = 45 Other strategies to consider: 1) Proportions – make it into a proportion and: a. Cross multiply if the bottoms of the fractions are different b. Clear the fractions if the bottoms of the fractions are the same Option a. Cross multiply if the bottoms of the fractions are different Example 1: 3x = 18 4 Turn it into 3x = 18 4 1 then cross multiply: (3x)(1) = 18*4  3x = 72 Example 2:  x = 24 3 = 18 4x Turn it into _3_ = 18 4x 1 then cross multiply: (3)(1) = 18(4 x)  3 = 72x Example 3: 3 + x = 18 4  x = 3/72 = 1/24 Turn it into _3 + x_ = 18 4 1 then cross multiply: (3 + x))(1) = 18(4 )  3 + x = 72  x = 69 Example 4: 3 + x = 18 - x 4 Turn it into _3 + x_ = 18 - x 4 1 then cross multiply: (3 + x))(1) = (4 )(18 – x)  3 + x = 72 – 4x  3 + 5x = 72  5x = 69  x = 69/5 Option b. Clear the fractions if the bottoms of the fractions are the same “Clearing the Fractions is a technique that a lot of students like, and GREATLY reduces their stress level!! To “Clear the Fractions”: 1) Get the bottom numbers the same for ALL numbers (including any whole numbers) 2) Cross out ALL the bottom numbers and re-write the equation  Fractions are GONE. The equations looks like a normal equation now. Revisiting Example 1 (above): 3x = 18 4 Re-write with 18 as a fraction, turning it into Make the bottoms the same: 3x = 18 4 1 3x = 18 4 1 becomes Cross out the bottoms and re-write: 3x = 72 4 4 3x = 72 4 4 becomes 3x = 72 Solve like normal: 3x = 72  x = 24  same answer as above Revisiting Example 2 (above): Turn it into 3 = 18 4x _3_ = 18 4x 1 Make the bottoms the same: 3 = 18 4x 1 becomes Cross out the bottoms and re-write: 3 = 72x 4x 4x 3 = 72x 4x 4x becomes 3 = 72x Solve like normal: 3 = 72x  x = 1/24  same answer as above Revisiting Example 3 (above): 3 + x = 18 4 Turn it into _3 + x_ = 18 4 1 Make the bottoms the same: 3 + x = 18 4 1 becomes 3 + x = 72 4 4 Cross out the bottoms and re-write: 3 + x = 72 becomes 3 + x = 72 4 4 Solve like normal: 3 + x = 72  x = 69  same answer as above Example 4: 3+x=6- x 2 5 4 Turn it into _3 + x = 6 - x 2 5 1 4 Make the bottoms the same  smallest number 2, 4, & 5 all go into is 20 3 + x = 6 – x becomes 2 5 1 4 30 + 4x = 120 – 5x 20 20 20 20 Cross out the bottoms and re-write: 30 + 4x = 120 – 5x becomes 30 + 4x = 120 – 5x 20 20 20 20 Solve like normal: 30 + 4x = 120 – 5x  9x = 90  x = 10  final answer Videos that may help: Order of Operations http://www.youtube.com/watch?v=_tHQgBt1KQ4 http://www.youtube.com/watch?v=WHTBgwBWlF4 Evaluate Algebraic Expressions http://www.youtube.com/watch?v=vQT0e_p_Z8s http://www.youtube.com/watch?v=WR-PI9303TE Combining Like Terms http://www.youtube.com/watch?v=C72lpuGQYo4 Simplifying Algebraic Expressions http://www.youtube.com/watch?v=_A8lLbZCrlw Solve Linear equations in one variable One-step Equations: The Basics http://www.youtube.com/watch?v=7T-_K_-VSJE Two-Step Equations http://www.youtube.com/watch?v=9ITsXICV2u0 Multiple-Step Equations http://www.youtube.com/watch?v=_PsDFffvBsI